Numerical Methods for Detecting Symmetries and Commutant Algebras

Abstract

For families of Hamiltonians defined by parts that are local, the most general definition of a symmetry algebra is the commutant algebra, i.e., the algebra of operators that commute with each local part. Thinking about symmetry algebras as commutant algebras allows for the treatment of conventional symmetries and unconventional symmetries (e.g., those responsible for weak ergodicity breaking phenomena) on equal algebraic footing. In this work, we discuss two methods for numerically constructing this commutant algebra starting from a family of Hamiltonians. First, we use the equivalence of this problem to that of simultaneous block-diagonalization of a given set of local operators, and discuss a probabilistic method that has been found to work with probability 1 for both Abelian and non-Abelian symmetries or commutant algebras. Second, we map this problem onto the problem of determining frustration-free ground states of certain Hamiltonians, and we use ideas from tensor network algorithms to efficiently solve this problem in one dimension. These numerical methods are useful in detecting standard and non-standard conserved quantities in families of Hamiltonians, which includes examples of regular symmetries, Hilbert space fragmentation, and quantum many-body scars, and we show many such examples. In addition, they are necessary for verifying several conjectures on the structure of the commutant algebras in these cases, which we have put forward in earlier works. Finally, we also discuss similar methods for the inverse problem of determining local operators with a given symmetry or commutant algebra, which connects to existing methods in the literature. A special case of this construction reduces to well-known ``Eigenstate to Hamiltonian" methods for constructing Hermitian local operators that have a given state as an eigenstate.

14+7 pages, 4 figures, v2: References added, Sec. V expanded

References in corpus (12)

• The density-matrix renormalization group in the age of matrix product states
• Probing many-body dynamics on a 51-atom quantum simulator
• Reference frames, superselection rules, and quantum information
• Controlling many-body dynamics with driven quantum scars in Rydberg atom arrays
• Eta-Pairing in Hubbard Models: From Spectrum Generating Algebras to Quantum Many-Body Scars
• Finite automata for caching in matrix product algorithms
• Eta-pairing states as true scars in an extended Hubbard Model
• Enlarged symmetry algebras of spin chains, loop models, and S-matrices
• Explicit construction of local conserved operators in disordered many-body systems
• Symmetry protected topological phases beyond groups: The q-deformed Affleck-Kennedy-Lieb-Tasaki model
• Numerical method for finding decoherence-free subspaces and its applications
• Coherence generation, symmetry algebras and Hilbert space fragmentation

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• Exact Model Reduction for Continuous-Time Open Quantum Dynamics
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• Non-Universality from Conserved Superoperators in Unitary Circuits
• Dynamics in the presence of local symmetry-breaking impurities
• Additional quantum many-body scars of the spin-$1$ $XY$ model with Fock-space cages and commutant algebras
• Absence of local conserved charges of the Fredkin spin chain and its truncated versions